Chern characters for supersymmetric field theories

TL;DR

This paper constructs a geometric bridge between supersymmetric field theories and advanced cohomology theories, providing new insights into Chern characters and modular forms within the Stolz--Teichner framework.

Contribution

It introduces a geometric construction linking supersymmetric field theories to K-theory and elliptic cohomology, advancing the Stolz--Teichner program and identifying models for Chern characters.

Findings

Established a map from 1|1-dimensional theories to K-theory.

Extended the construction to 2|1-dimensional theories and elliptic cohomology.

Provided a geometric proof that certain partition functions are weak modular forms.

Abstract

We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parameterized generalization of Fei Han's realization of the Chern character in K-theory as dimensional reduction for $1|1$-dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2|1$-dimensional tori and the derived geometry of complex analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $\mathcal{N}=(0,1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.